## The Annals of Probability

### A variational approach to dissipative SPDEs with singular drift

#### Abstract

We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which neither continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques with a priori estimates, both pathwise and in expectation, on solutions to regularized equations.

#### Article information

Source
Ann. Probab., Volume 46, Number 3 (2018), 1455-1497.

Dates
Revised: March 2017
First available in Project Euclid: 12 April 2018

https://projecteuclid.org/euclid.aop/1523520022

Digital Object Identifier
doi:10.1214/17-AOP1207

Mathematical Reviews number (MathSciNet)
MR3785593

Zentralblatt MATH identifier
06894779

#### Citation

Marinelli, Carlo; Scarpa, Luca. A variational approach to dissipative SPDEs with singular drift. Ann. Probab. 46 (2018), no. 3, 1455--1497. doi:10.1214/17-AOP1207. https://projecteuclid.org/euclid.aop/1523520022

#### References

• [1] Albeverio, S., Kawabi, H. and Röckner, M. (2012). Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions. J. Funct. Anal. 262 602–638.
• [2] Arendt, W. (2006). Heat Kernels, Lecture Notes of the 9th Internet Seminar on Evolution Equations. Available at https://www.uni-ulm.de/mawi/iaa/members/arendt/.
• [3] Arendt, W. and Bukhvalov, A. V. (1994). Integral representations of resolvents and semigroups. Forum Math. 6 111–135.
• [4] Arendt, W., Chill, R., Seifert, C., Vogt, D. and Voigt, J. (2015). Form methods for evolution equations and applications. In Lecture Notes of the 18th Internet Seminar on Evolution Equations. Available at https://www.mat.tuhh.de/isem18/Phase_1:_The_lectures.
• [5] Barbu, V. (1993). Analysis and Control of Nonlinear Infinite-Dimensional Systems. Academic Press, Boston, MA.
• [6] Barbu, V. (2010). Existence for semilinear parabolic stochastic equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 21 397–403.
• [7] Barbu, V., Da Prato, G. and Röckner, M. (2009). Existence of strong solutions for stochastic porous media equation under general monotonicity conditions. Ann. Probab. 37 428–452.
• [8] Barbu, V. and Marinelli, C. (2009). Strong solutions for stochastic porous media equations with jumps. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 413–426.
• [9] Bendikov, A. and Maheux, P. (2007). Nash type inequalities for fractional powers of non-negative self-adjoint operators. Trans. Amer. Math. Soc. 359 3085–3097.
• [10] Boccardo, L. and Croce, G. (2014). Elliptic Partial Differential Equations. De Gruyter, Berlin.
• [11] Bourbaki, N. (1981). Espaces Vectoriels Topologiques. Chapitres 1 à 5, New ed. Masson, Paris.
• [12] Brézis, H. (1973). Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam.
• [13] Brézis, H. (1971). Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. In Contributions to Nonlinear Functional Analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) 101–156. Academic Press, New York.
• [14] Brzeźniak, Z., Maslowski, B. and Seidler, J. (2005). Stochastic nonlinear beam equations. Probab. Theory Related Fields 132 119–149.
• [15] Cerrai, S. (2003). Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields 125 271–304.
• [16] Dautray, R. and Lions, J.-L. (1988). Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 2. Springer, Berlin.
• [17] Davies, E. B. (1990). Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge.
• [18] Da Prato, G. (2004). Kolmogorov Equations for Stochastic PDES. Birkhäuser, Basel.
• [19] Engel, K.-J. and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194. Springer, New York.
• [20] Gentil, I. and Imbert, C. (2008). The Lévy–Fokker–Planck equation: $\Phi$-entropies and convergence to equilibrium. Asymptot. Anal. 59 125–138.
• [21] Haase, M. (2007). Convexity inequalities for positive operators. Positivity 11 57–68.
• [22] Hiriart-Urruty, J.-B. and Lemaréchal, C. (2001). Fundamentals of Convex Analysis. Springer, Berlin.
• [23] Kato, T. (1995). Perturbation Theory for Linear Operators. Springer, Berlin. Reprint of the 1980 edition.
• [24] Krylov, N. V. and Rozovskiĭ, B. L. (1979). Stochastic evolution equations. In Current Problems in Mathematics, Vol. 14 (Russian) 71–147, 256. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow.
• [25] Kunze, M. and van Neerven, J. (2012). Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations. J. Differential Equations 253 1036–1068.
• [26] Kusuoka, S. and Marinelli, C. (2014). On smoothing properties of transition semigroups associated to a class of SDEs with jumps. Ann. Inst. Henri Poincaré Probab. Stat. 50 1347–1370.
• [27] Lions, J.-L. and Magenes, E. (1968). Problèmes aux Limites Non Homogènes et Applications. Vol. 1. Dunod, Paris.
• [28] Liu, W. and Röckner, M. (2015). Stochastic Partial Differential Equations: An Introduction. Springer, Cham.
• [29] Ma, Z. M. and Röckner, M. (1992). Introduction to the Theory of (Nonsymmetric) Dirichlet Forms. Springer, Berlin.
• [30] Marinelli, C. On well-posedness of semilinear stochastic evolution equations on $L_{p}$ spaces. Preprint. Available at arXiv:1512.04323.
• [31] Marinelli, C. and Quer-Sardanyons, L. (2012). Existence of weak solutions for a class of semilinear stochastic wave equations. SIAM J. Math. Anal. 44 906–925.
• [32] Ouhabaz, E. M. (2005). Analysis of Heat Equations on Domains. Princeton Univ. Press, Princeton, NJ.
• [33] Pardoux, É. (1975). Equations aux derivées partielles stochastiques nonlinéaires monotones, Ph.D. Thesis, Univ. Paris XI.
• [34] Pardoux, E. and Răşcanu, A. (2014). Stochastic Differential Equations, Backward SDES, Partial Differential Equations. Springer, Cham.
• [35] Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York.
• [36] Simon, J. (1987). Compact sets in the space $L^{p}(0,T;B)$. Ann. Mat. Pura Appl. (4) 146 65–96.
• [37] Strauss, W. A. (1966). On continuity of functions with values in various Banach spaces. Pacific J. Math. 19 543–551.
• [38] van Neerven, J. M. A. M., Veraar, M. C. and Weis, L. (2008). Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal. 255 940–993.
• [39] Varopoulos, N. Th., Saloff-Coste, L. and Coulhon, T. (1992). Analysis and Geometry on Groups. Cambridge Univ. Press, Cambridge.